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\pub{2009}{1}{3}{1}
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\topic{Lecture 2 \\Differential Calculus-I\\ \scriptsize Leibnitz's Theorem (16 Sep 2009)}
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This theorem is useful for finding the $n$th differential coefficient of a product. It is as follows:
\begin{quote}
If $u$ and $v$ are two functions of $x$, then
\[D^n(uv) = ^nC_0 D^nu.v + ^nC_1 D^{n-1}u.Dv+^{n-2}C_2 D^{n-2}u.D^2v+ ... + u.^nC_n D^nv\]
\end{quote}
We shall prove this theorem by mathematical induction. Consider $n=1$, i.e. $D(uv)$, By actual differentiation we have
\[D(uv) = Du.v + u.Dv\]
which shows the theorem is true for $n=1$. Now assume that the result is true for a particular value of $n$, say $n=m$, then we have
\[D^m(uv) = ^mC_0 D^mu.v + ^mC_1 D^{m-1}u.Dv+^{m-2}C_2 D^{m-2}u.D^2v+ ... + u.^mC_m D^mv\]
Then differentiating on both side with respect to $x$, we get
\[
\begin{array}{rcl}
D^{m+1}(uv) &=& ^mC_0 [D^{m+1}u.v+D^mu.Dv] \\
 &+& ^mC_1 [D^{m}u.Dv+D^{m-1}u.D^2v]\\
 &+& ^mC_2 [D^{m-1}u.D^2v+D^{m-2}u.D^3v]\\
 &+& ...\\
 &+& ^mC_m [Du.D^mv+u.D^{m+1}v]\\
\end{array}
\]
On rearranging,and using $^nC_r + ^{n}C_{r+1} = ^{n+1}C_{r+1}$, we have
\[D^{m+1}(uv) = ^{m+1}C_0 D^{m+1}u.v + ^{m+1}C_1 D^{{m+1}-1}u.Dv+^{{m+1}-2}C_2 D^{{m+1}-2}u.D^2v+ ... + u.^{m+1}C_{m+1} D^{m+1}v\]
This implies that if theorem is true for $n=m$, it is true for the next value of $n$. It also holds for $n=1$, hence it must be true for every positive integral value of $n$.
\begin{example}
Find the $n$th differential coefficient of $x^3e^{ax}$.
\end{example}
Let $v=x^3$, and $u=e^{ax}$, Now,
\[
\begin{array}{rcl}
D^n(x^3e^{ax}) &=& D^ne^{ax}.x^3 + ^nC_1 D^{n-1}e^{ax}.Dx^3+^{n}C_2 D^{n-2}e^{ax}.D^2x^3+^{n}C_3 D^{n-3}e^{ax}.D^3x^3\\
 &=& a^ne^{ax}.x^3 + n a^{n-1}e^{ax}.3x^2+ \frac{n(n-1)}{2} a^{n-2}e^{ax}.6x+^{n}C_3 a^{n-3}e^{ax}.6\\
\end{array}
\]
Hence
\begin{example}
Differentiate $n$ times the equation
\[(1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}+a^2y = 0\]
\end{example}
We may write the equation as
\[(1-x^2)y_2-xy_1+a^2y = 0\]
\[
\begin{array}{lcl}
D^n[(1-x^2)y_2] &=& (1-x^2)y_{n+2}+n(-2x)y_{n+1}+\frac{n(n-1)}{2}(-2)y_n\\
D^n[-xy_1] &=& -xy_{n+1}-n(1)y_{n}\\
D^n[a^2y] &=& a^2y_{n}\\
\end{array}
\]
On adding all these, we get
\[D^n[(1-x^2)y_2-xy_1+a^2y]=(1-x^2)y_{n+2}-(2n+1)xy_{n+1}-(n^2-a^2)y_n\]
\[(1-x^2)y_{n+2}-(2n+1)xy_{n+1}-(n^2-a^2)y_n = 0\]
as $(1-x^2)y_2-xy_1+a^2y = 0$\\
Hence
\section*{Problems}
\begin{enumerate}
\item   Apply Leibnitz's Theorem to find yn in the following cases:\\
				(a) $x^3 e^{ax}$~~	(b)	$x^3 \cos x$~~    (c) $x^4 \sin 2x$ ~~(d) $x^3 \log x$~~ 	(e)	$x^2 e^x \cos x$

\item   If $y= x^2 \sin x$, prove that $\frac{d^{n} y}{dx^{n} } =\left(x^{2} -n^{2} +n\right)\sin \left(x+\frac{n\pi }{2} \right)-2nx\cos \left(x+\frac{n\pi }{2} \right)$ 

\item   If $y^{1/m} + y^{-1/m} = 2x$, prove that \[(x^2 -1) y_{n+2} + (2n+1)x y_{n+1} + (n^2 - m^2) y_n = 0\]

\item   If $y = a \cos (\log x) + b \sin (\log x)$, show that 
(a)	$x^2 y_2 + x y_1 + y = 0$ and (b)	$x^2  y_{n+2} + (2n+1)x y_{n+1} + (n^2+1) y_n = 0$.

\item   If $y = (x^2 -1)^n$, prove that \[(x^2 -1) y_{n + 2} + 2x y_{n + 1} - n (n + 1)y_n = 0\]

\item   If $\cos ^{-1} \left(\frac{y}{b} \right) = \log \left(\frac{x}{m} \right)^{m}$, prove that \[x^2y_{n + 2} + (2n+ 1) x y_{n + 1} + (n^2 + m^2) y_n = 0\]

\item  If $\cosh \left(\frac{1}{m} \log y\right)$ prove that $(x^2 - 1) y_{n + 2} + (2n+ 1) x y_{n + 1} + (n^2-m^2)y_n = 0$.

\item   If $y =\left(1-x\right)^{-\alpha } e^{-\alpha x} $, prove that $(1-x) y_{n + 1}- (n+\alpha x)y_{n} -n\alpha y_{n-1} =0$. 

\item  If \textit{y} = $\log \left(x+\sqrt{1+x^{2} } \right)$ , prove that \[(1 + x^2) y_{n + 2} + (2n + 1)xy_{n+1} + n^2y_n = 0\]  

\item   If $x = \tan (\log y)$, prove that $(1 + x^2) y_{n+1} + (2nx-1) y_n + n (n-1) y_{n-1} = 0$. 
\item  If $y = \sin \log_e (x^2 + 2x + 1)$, prove that $(1 + x^2) y_{n+2} + (2n + 1) (1+x) y_{n+1} + (n^2 + 4) y_n = 0$.

\item   If $y = x^n \log x$ prove that 
(a) $y_{n + 1}= n! /x$ (b)  $y_n = n y_{n -1} + (n-1)!$ (c) Show that $D^n (x^{n-1} \logx) = \frac{(n-1)!}{x}$.

\item   If $y = (sin^{-1}x)^2$, prove that 
(a) $(1-x^2) \frac{d^{2} y}{dx^{2}} - x\frac{dy}{dx} -2=0$  (b) $(1-x^2) y_{n + 2} - (2n + 1) xy_{n + 1} - n^2y_n = 0$.



\item   If $y=  \sin^{-1} x$, prove that $(1-x^2) y_{n+2} - (2n+1) xy_{n+1}-n^2 y_n = 0$

\item   If $y = e^{m\sin ^{-1} x} $, prove that: 
(a) 	$(1-x^2) y_2 -x y_1 = m^2y$ (b)  $(1-x^2) y_{n+2} - (2n+1) xy_{n+1} - (n^2+m^2) y_n = 0$

\item  If $x=\sin \left(\frac{\log y}{a} \right)$, prove that $(1-x^2) y_{n+2} - (2n+1)x y_{n+1} - (n^2+a^2) y_n = 0$

\item   IF $y=\frac{\sin ^{-1} x}{\sqrt{1-x^{2} } } $, prove that  $(1-x^2) y_{n + 2} - (2n + 3) x y_{n + 1} - (n + 1)^2 y_n = 0$

\item   If $y=e^{\tan ^{-1} x} $, prove that $(1-x^2) y_{n + 2}- [(2n + 2) x-1] y_{n+1} + n (n + 1) y_n = 0$

\item   If $y = \cos(\log x)$ , prove that  $x^2 y_{n+2} + (2n+1) x y_{n+1} + (n^2+1) y_n = 0$

\item   If $y = x \cos(\log x)$, then show that  $x^2 y_{n+2} + (2n-1) x y_{n+1} + (n^2-2n+2) y_n = 0$

\item   If $y = (\sin^{-1}x)^2$ , prove that 
(a) $(1-x)^2 y_2 - x y_1 - 2 = 0$  (b) $(1-x^2) y_{n+2} - (2n+1) x y_{n+1} - n^2 y_n = 0$

\item   If $y = (\tan^{-1}x)^2$ , prove that $(x^2+1)^2 y_2 + 2x(1+x^2) y_1 = 2$

\item   If $\cos ^{-1} \left(\frac{y}{b} \right)=\log \left(\frac{x}{n} \right)^{n} $, prove that $x^2 y_{n+2} + (2n+1) x y_{n+1} + 2n^2y_n = 0$

\item   If $y = \cos (m \sin^{-1}x)$,  prove that  $(1- x^2) y_{n + 2} - (2n + 1) x y_{n + 1} - (n^2 - a^2) y_n = 0$

\item   If $y = \sin (m \sinh^{-1}x)$,  prove that  $(1+ x^2) y_{n + 2} + (2n + 1) x y_{n + 1} + (n^2 + m^2) y_n = 0$

\item   If $y = (\sinh^{-1}x)^2$ , prove that  $(1+x^2) y_{n + 2} + (2n+1) x y_{n+1} + n^2 y_n = 0$

\item   If $y = \sqrt{1-x^{2}} \sin^{-1} x$, prove that 
(a) $y_3 (1-x^2) -3y_2 x + 2 = 0$ (b) $y_{n + 3} (1- x^2) - (2n + 3) xy_{n + 2} -n(n + 2) y_{n + 1} = 0$
\item   Prove that \[\frac{d^{n} }{dx^{n} } \left(\frac{\sin x}{x} \right)=\left[P\sin \left(x+\frac{n\pi }{2} \right)+Q\cos \left(x+\frac{n\pi }{2} \right)\right]/x^{n+1} \] where \[P = x^n - n(n-1)x^{n - 2} + n(n-1)(n-2)(n-3)x^{n -4} - \dots\]  and \[Q = nx^{n-1} - n(n-1)(n-2) x^{n-3} + \dots\]

\item   If $y =\frac{\log x}{x} $, prove that $y_n = \frac{(-1)^{n} n!}{x^{n+1} } \left(\log x - 1 - \frac{1}{2}-\frac{1}{3}- \dots -\frac{1}{n}\right)$.

\item   If $x + y = 1$, prove that 
\[\frac{d^{n} }{dx^{n} } \left(x^{n} y^{n} \right)=n!\left[y^{n} -\left({}^{n} c_{1} \right)^{2} y^{n-1} .x+\left({}^{n} c_{2} \right)^{2} y^{n-2} .x^{2} -....+\left(-1\right)^{n} x^{n} \right]\]
\item   If $x\sin \phi +y\cos \phi =a$ and $x\cos \phi -y\sin \phi =b$ show that \[\frac{d^{m} x}{d\phi ^{m}} .\frac{d^{n} y}{d\phi ^{n} } -\frac{d^{m} y}{d\phi ^{m} } .\frac{d^{n} x}{d\phi ^{n} } = Constant\]

\item   If $y = A(x + \sqrt{x^{2} +a^{2} })^n + B(x +\sqrt{x^{2} +a^{2} } )^{-n}$, prove that 
\[(a^2+x^2) y_{m+2} + (2m+1) x y_{m+1} + (m^2-n^2) y_m = 0\].

\item   Evaluate $D^n [\log (x^2-2x\cos \alpha + 1)]$ hence or otherwise find $D^n \left[\tan ^{-1} \left(\frac{x\sin \alpha }{1-x\cos \alpha } \right)\right]$.

\item   Prove that $1+\frac{n^{2} }{1^{2} } +\frac{n^{2} (n-1)^{2} }{1^{2} .2^{2} } +\frac{n^{2} (n-1)^{2} (n-2)^{2} }{1^{2} .2^{2} .3^{2} } +....=\frac{2n!}{(n!)^{2} }$.

\item   If $I_n =\frac{d^{n} }{dx^{n} } (x^{n} \log x)$, prove that $I_n = n!\left[\log x+1+\frac{1}{2} +\frac{1}{3} +.....+\frac{1}{n} \right]$.

\item   If $y = A (x + \sqrt{x^{2} -1} )^n + B(x - \sqrt{x^2-1} )^n$ , prove that 
(a) $(x^2-1) y_2 + xy_1 - n^2y = 0$
(b) $(x^2-1) y_{n+2} + (2n+1) xy_{n+1}  = 0$

\item   If $y \sqrt{x^{2} -1} =\log (x+\sqrt{x^{2} -1} )$, show that \[(x^2-1) y_{n+1} + (2n+1) x y_n +n^2 y_{n-1} = 0\]

%\item   If $y=  sin^{-1} x$, prove that $(1-x^2) y_{n+2} - (2n+1) x y_{n+1}-n^2 y_n = 0$ Also find the value of $y_n$ when $x = 0$.
%
%\item   If$y=\frac{\sin ^{-1} x}{\sqrt{1-x^{2} } } $, prove that $y_n = (n-1)^2 y_{n-2}$ for $x = 0$.
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%\item   If $y =[x+\sqrt{1+x^{2} } ]^{m} $, find $y_n(0)$.
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%\item   Find $y_n (0)$ when $y = \log(x+\sqrt{1+x^{2}})$.
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%\item   If $y = [\log(x+\sqrt{1+x^{2} } )]^2$, find all the derivatives of $y$ w.r.t. $x$, when $x = 0$. 
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%\item  If $y = (sinh^{-1}x)^2$, prove that $(1+x^2) y_{n+2} + (2n+1) x y_{n+1} + n^2 y_n = 0$. Hence find $y_n$ when $x = 0$.
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%\item   If $y = e^{a\sin ^{-1} x} $, prove that $(1-x^2) y_{n+2} - (2n+1) xy_{n+1} - (n^2+a^2) y_n = 0$. Deduce that $\mathop{Lt}\limits_{x\to 0} $$\frac{y_{n+2} }{y_{n} }  = n^2 + a^2$. Hence find $y_n (0)$.
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%\item   If $y = \sin (m \sin^{-1}x)$, find $y_n(0)$.
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%\item   Prove that the value when $x = 0$ of $D^n(tan^{-1}x)$ is $0,(n-1)!$ Or $-(n-1)!$  according as $n$ is of the form $2p, 4p + 1$ or $4p + 3$ respectively.
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%\item   If $y = \log (x+\sqrt{x^{2} +a^{2} })$, prove that $(a^2 + x^2) y_2 + xy_1 = 0$. Differentiate this equation $n$ times and prove that $\mathop{\lim }\limits_{x\to 0} \frac{y_{n+2} }{y_{n} } =-\frac{n^{2} }{a^{2} } $.
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%\item   If $\log y = tan^{-1}x$, show that \[(1+ x^2) y_{n + 2} + \{2(n + 1) x - 1\} y_{n + 1} + n (n+1) y_n = 0\] hence find $y_3, y_4$ and $y_5$ at $x = 0$.
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%\item   If $y = e^{m \cos ^{-1} x}$, show that $(1- x^2) y_{n + 2}- (2n + 1) x y_{n + 1} - (n^2 + m^2) y_n = 0$ and calculate $y_n(0)$.
%\item   If $y = \tan^{-1}x$, prove that $(1+ x^2) y_{n + 1} + 2nxy_n + n (n-1) y_{n - 1} = 0$. Hence determine the values of all the derivatives of $y$ with respect to $x$ when $x =0$. \\\textbf{Ans}: When $n$ is even, $y_n(0) = 0$. When $n$ is odd, $y_n(0) = (-1)(n-1)/2(n-1)! $ ???
\end{enumerate}

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